To say that recent activity in the US Equity market has been unprecedented would certainly be an understatement. On Friday, August 21st, and Monday, August 24th the S&P 500 index fell 3.24% and 4.02% respectively. To put these declines in perspective: based on a year's worth of data, Monday's market drop was equivalent to an over 4 standard deviation event, making these declines extreme by any statistical measure:

Date

Return

Standard Deviations

August 20

-2.13%

2.86

August 21

-3.24%

4.17

August 24

-4.02%

4.83

For some perspective, to correctly estimate last Friday's decline using a year of daily data through August 20th, a parametric VaR would require a confidence interval of 99.9984%. This equates to the probability of such an event occurring to be less than once in 10,000 observations.

On the subject of VaR, one would require using data going back to 2008 to anticipate the returns just seen on Friday, August 21st:

For Monday, August 24th's decline of 77.68 points, only a conditional VaR using 7-years of data (again, including 2008) correctly estimated the S&P 500's decline that day:

Which brings us to why stress-testing is exceedingly important as a complimentary set of exposure analysis. Here we shock an at-the-money option on the SPY's starting with data as of August 20th using a -5% move in the S&P 500 Index:

The benefit of stress-testing is its lack of reliance on statistical (historical) data. Regardless of the presence of extreme events in a given data set (or detrimentally, in this case, the lack thereof), stress-testing allows for simulation of market shocks in all environments, volatile, extreme, or not.

Earlier this month saw volatility return to nearly every tradable market with a serious vengeance. The long-running low volatility (or no volatility, to some) state of affairs had apparently come to an end. For those of us that construct and test risk systems, this month has provided a unique in situ series of observations with which to critically examine the performance of certain VaR models.

Defining the low-vol environment

With regards to equity markets, at least, it will certainly come as no surprise that volatility has been low and getting lower for quite some time. Below is a chart of rolling 90-day S&P500 realized volatility starting on Jan 1 of this year through September 24th (one day prior to the first large down moves seen in the US Equity markets starting on 9/25/2014):

Even taking the period's peak volatility of ~ 13%, these are historically very low volatility levels for the US Equity markets. For the purposes of this evaluation we will consider the "pre-volatility" period to be from Jan 1-September 24th of 2014.

Defining the volatility-inducing events

Starting on September 25th, the S&P500 experienced a series of several outsized, negative returns:

Date

Return %

Return Points

09/25/2014

-1.63%

-32.31

10/01/2014

-1.33%

-26.13

10/07/2014

-1.52%

-29.72

10/09/2014

-2.09%

-40.68

10/10/2014

-1.15%

-22.08

10/13/2014

-1.66%

-31.39

These were certainly enough observations to make even the most ardent believer in the "endless-low-volatility-bull-market" theory question his or her convictions.

Which VaR model did best?

Using data from January 1 to September 24th, we ran the S&P500 through the Parametric, Historical Simulation, and Monte Carlo VaR models offered in RiskAPI, both under 95% and 99% confidence levels. Here are the results (in index points):

Model

Confidence

VaR

CVaR

Parametric

95%

-20.69

-31.19

Parametric

99%

-29.27

-39.34

Full Historical

95%

-18.56

-28.84

Full Historical

99%

-39.13

-40.05

Monte Carlo

95%

-22.21

-27.00

Monte Carlo

99%

-31.29

-38.00

The last two columns in the results table above show the output of Value at Risk (VaR) and Conditional Value at Risk (CVaR) across each of the three models and two confidence levels. While VaR shows the predicted return event based on a statistical model, CVaR shows how large the actual historical period losses that exceed this prediction were, on average.

By far, the worst day for the S&P this month was on October 9th, with a loss over 40 points, or more than 2%. As a risk manager, being able to forecast such a loss well-prior to experiencing it would clearly be advantageous. At the 95% confidence level, no model using either VaR or CVaR came close to predicting the loss on October 9th. However, this is more of a function of how rare this event was than the efficacy of 95% VaR. Remember that we fully expect a 95% VaR result to fall short 5% of the time.

Using a 99% confidence level, the Full Historical model came quite close to forecasting this event using the "low volatility" data alone, while CVaR did very well across all models. The performance of CVaR in this situation speaks very highly of this metric, showing that even with event-limited data, one is able to predict large losses by explicitly examining behavior at the "tail" loss region of a distribution.

For completeness, here is what realized volatility looks like taking events since September 24th into account: